JEE Advance - Physics (2013 - Paper 1 Offline - No. 10)
Two non-conducting solid spheres of radii $$R$$ and $$2R,$$ having uniform volume charge densities $${\rho _1}$$ and $${\rho _2}$$ respectively, touch each other. The net electric field at a distance $$2$$ $$R$$ from the center of the smaller sphere, along the line joining the centers of the spheres, is zero. The ratio $${{{\rho _1}} \over {{\rho _2}}}$$ can be
$$-4$$
$$ - {{32} \over {25}}$$
$$ {{32} \over {25}}$$
$$4$$
Explanation
The electric field at $Q$
$$ \vec{E}_Q=\frac{1}{4 \pi \epsilon_0}\left[\frac{\frac{4}{3} \pi R^3 \rho_1}{(2 R)^2} \hat{\imath}-\frac{\frac{4}{3} \pi R^3 \rho_2}{R^2} \hat{\imath}\right]=\overrightarrow{0} $$
gives $\rho_1 / \rho_2=4$
The electric field at $P$,
$$ \vec{E}_P=-\frac{1}{4 \pi \epsilon_0}\left[\frac{\frac{4}{3} \pi R^3 \rho_1}{(2 R)^2} \hat{\imath}+\frac{\frac{4}{3} \pi(2 R)^3 \rho_2}{(5 R)^2} \hat{\imath}\right]=\overrightarrow{0}, $$
gives $\rho_1 / \rho_2=-32 / 25$.
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